Several known heuristic functions can capture the input at different levels of precision, and support relaxation-refinement operations guaranteeing to converge to exact information in a finite number of steps. A natural idea is to use such refinement online, during search, yet this has barely been addressed. We do so here for local search, where relaxation refinement is particularly appealing: escape local minima not by search, but by removing them from the search surface. Thanks to convergence, such an escape is always possible. We design a family of hill-climbing algorithms along these lines. We show that these are complete, even when using helpful actions pruning. Using them with the partial delete relaxation heuristic h~(CFF), the best-performing variant outclasses FF's enforced hill-climbing, outperforms FF, outperforms dual-queue greedy best-first search with h~(FF), and in 6 IPC domains outperforms both LAMA and Mercury.
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